Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies

Abstract

We first seek the rebit-retrit counterpart to the (formally proven by Lovas and Andai) two-rebit Hilbert-Schmidt separability probability of $\frac{29}{64}=\frac{29}{2^6}\approx 0.453125$ and the qubit-qutrit analogue of the (strongly supported) value of $\frac{8}{33}=\frac{2^3}{3\cdot 11}\approx 0.242424$. We advance the possibilities of a rebit-retrit value of $\frac{860}{6561}=\frac{2^2\cdot 5\cdot 43}{3^8}\approx 0.131078$ and a qubit-qutrit one of $\frac{27}{1000}=(\frac{3}{10}{)}^3=\frac{3^3}{2^3\cdot 5^3}=0.027$. These four values for $2\times m$ systems ($m=2,3$) suggest certain numerator/denominator sequences involving powers of $m$, which we further investigate for $m>3$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3{\pi }^2}\approx 0.54038$, as well as more generally, that the probabilities based on induced measures are equal across these three sets. Then, we extend the master Lovas-Andai formula to induced measures. For instance, the two-qubit function ($k=0$) is ${\tilde{\chi }}_{2,0}(\epsilon )=\frac{1}{3}{\epsilon }^2(4-{\epsilon }^2)$, yielding $\frac{8}{33}$, while its $k=1$ induced measure counterpart is ${\tilde{\chi }}_{2,1}(\epsilon )=\frac{1}{4}{\epsilon }^2{\left(3-{\epsilon }^2\right)}^2$, yielding $\frac{61}{143}=\frac{61}{11\cdot 13}\approx 0.426573$, where $\epsilon$ is a singular-value ratio. Interpolations between Hilbert-Schmidt and operator monotone (Bures, $\sqrt{x}$) measures are also studied. Using a recently-developed golden-ratio-related (quasirandom sequence) approach, current (significant digits) estimates of the two-rebit and two-qubit Bures separability probabilities are 0.15709 and 0.07331, respectively.

52 pages, 7 figures, 3 appendices of Charles Dunkl, includes much of arXiv:1803.10680

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Cited by in corpus (3)

• Quantitative characterization of several entanglement detection criteria
• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities